**PhilMath Intersem** is jointly sponsored by the History and Philosophy of Science department of the University of Paris 7-Diderot, the University of Notre Dame and the research unit SPHERE (UMR7219) of the University of Paris Diderot

The 9th edition of PhilMath Intersem will take place in June 2018.

The theme of this year’s seminar will be the use of non-deductive methods in mathematics.

The language of the seminar will be English.

Abstracts, advices of reading are available underneath and on this website: https://mdetlefsen.nd.edu/philmath-intersem/philmath-intersem-9-2018/

Contact: Emmylou Haffner (U. Wuppertal & SPHERE)

PROGRAMME 2017-2018

Toutes les séances auront lieu salle Klimt (366A) du bâtiment Condorcet (10 rue Alice Domon & Léonie Duquet, 75013 Paris), sur le campus de l’Université Paris Diderot.

Theme 2018 : **use of non-deductive methods in mathematics.**

Sessions of 5, 7, 11, 14, 19, 21, 26.

- 14:00
Catherine Goldstein (CNRS, Institut de mathématiques de Jussieu-Paris Rive Gauche)
*Baconian mathematics*

It is usual to oppose mathematics to the so-called Baconian, that is experimental, sciences. Thomas Kuhn even suggested that modern science was created by the separation of experimental science from mathematical sciences during the seventeenth century. However, a number of early-modern mathematicians shared Francis Bacon’s criticism of deductive procedures, either syllogistic or Euclidean. Focusing on Marin Mersenne’s circle, in particular the work of Bernard Frenicle de Bessy (a founding member of the French Academy of Sciences), I shall discuss how some of the mathematical production in this circle went even further, following closely Baconian precepts. My key questions will be how induction and observation are used to produce mathematics, and in particular which type of induction for which mathematics.

- 16:00 Eberhard Knobloch (Institut für Philosophie, Literatur-, Wissenschafts- und Technikgeschichte, Berlin)
*Archimedes and his successors Kepler and Leibniz: Their use of analogies and equivalences in mathematics*.

Archimedes used analogies in the context of discovery. Especially interesting examples are his calculation of the surface of the sphere and his way of transferring the structure of quantities to the structure of non-quantities or indivisibles. Kepler amply used analogies in his "New solid geometry of wine barrels“ (1615) also in the context of justification referring to Archimedes without knowing his famous letter to Eratosthenes (so-called Method). Leibniz, too, referred to Archimedes in order to justify his infinitesimal calculus. He considered analogies as special harmonies that are closely connected to his law of continuity. He transferred the structure and laws of finite quantities to that of infinitely small and infinite quantities. Curves have to be considered as equivalent to a polygon of infinitely many sides. An instructive example is his geometrical integration of the logarithmic curve. Mathematical creativity is closely connected to the surpassing of limits.

- 16:00
James Franklin (School of mathematics and statistics, University of New South Wales, Sydney)
*Evidence for mathematical conjectures and non-deductive logic.*

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as the Riemann Hypothesis, have to be considered in terms of the evidence for and against them in the same way as legal verdicts and scientific hypotheses. After showing some examples, it is argued that it is not adequate to describe the relation of evidence to hypothesis in these cases as "subjective", "heuristic" or "pragmatic", but that it has an element of what it is strictly rational to believe on the evidence, that is, of non-deductive logic (or logical probability or objective Bayesianism with imprecise probabilities).

- 14:00
Jean Paul Van Bendegem (Centrum voor Logica en Wetenschapsfilosofie, Center for Logic and Philosophy of Science, Vrije Universiteit, Brussells)
*Can there be experiments, real or imaginary, in mathematics?*

At first sight the idea of a mathematical experiment seems an impossibility. Should mathematics, seen as the deductive discipline par excellence, not be ‘immune’ from empirical and experimental considerations? In Van Bendegem (1998) I firmly believed this to be the case but in Van Bendegem (1996) I equally firmly changed my mind on the issue (incidentally, the order of publication is the inverse of the order of writing) and defended the possibility of (particular types of) mathematical experiments. Faced with Starikova & Giaquinto (2018) on the specific issue of thought experiments and given the rapid development of so-called ‘experimental mathematics’, a re-evaluation of the issue seems appropriate. The conclusion is that mathematical experiments are indeed a genuine possibility but that a single definition of what they are seems unlikely, given the heterogeneity of cases.

- 16:00
Emmylou Haffner (Bergische Universität Wuppertal & SPHERE)
*On the computations underlying (some of?) Dedekind’s “conceptual mathematics”.*

Richard Dedekind is usually presented as one of the founding fathers, with Bernhard Riemann, of so-called conceptual mathematics. Both advocated for the importance of grounding a theory on concepts rather than computations. Computations, they wrote, should follow from rather than lead the definitions of the fundamental notions of a theory. Yet, if one looks more closely into their Nachlässe, one notices the remarkable quantity of computations on which their preliminary researches rely. In this talk, I will concentrate on Richard Dedekind’s mathematical works, and propose to dive into some of his manuscripts so as to understand how these computations were for him a stepwise way to come to more general (“conceptual”) algebraic laws. I will consider, in particular, the researches that let to his papers on Dualgruppen (similar to what is today called lattices) (Dedekind 1897, 1900). An analysis of his drafts suggests that Dedekind’s statements concerned the way one should write and present a theory to the scientific community and for its further uses, rather than the ways in which one investigates and elaborates new mathematical ideas properly speaking. I will argue that in the latter, the mathematics “in the making” so to speak, mathematical practices take on different forms and are closer to mathematical exploration and experimentation through computations and gradual generalization.

- 16:00
James Franklin (School of mathematics and statistics, University of New South Wales, Sydney)
*The nature of non-deductive logic in the light of its use in mathematics.*

The fact that non-deductive/probabilistic methods work in pure mathematics shows they are fully logical. For example, since induction works in pure mathematics, it cannot depend on any contingent fact such as the "uniformity of nature". It is argued that non-deductive methods are as strictly logic as deductive logic, as in Keynes’ view that logical probability is a degree of partial implication. Philosophies of the nature of logic are canvassed, in the light of which non-deductive inference forms like the confirmation of theories by their consequences and the proportional syllogism are exhibited as purely logical. Non-deductive logic is no less formal than deductive logic.

- 14:00
Alex Paseau (Faculty of Philosophy, Wadham College, University of Oxford)
*Knowledge of Mathematics without Proof*

Mathematicians do not claim to know a proposition unless they think they possess a proof of it. For all their confidence in the truth of a proposition with weighty nondeductive support, they maintain that, strictly speaking, the proposition remains unknown until such time as someone has proved it. My talk will present and further develop some arguments in my 2015 BJPS paper of the same title that challenge this conception of knowledge within mathematics.

- 16:00 Karine Chemla (CNRS, SPHERE, University of Paris Diderot)
*When a diagram writes a proof*

The talk will focus on a specific diagram, which, I claim, writes the correctness of an algorithm solving a quadratic equation numerically. After having outlined elements of context that are required to interpret the diagram, I will analyze the techniques used to formulate the proof in this way. We will thus see how the meaning of a single diagram unfolds while reading the text of the algorithm.

- 16:00
Christine Proust & Adeline Reynaud (SPHERE, Université Paris 7-Diderot)
*Reasoning with diagrams and layouts, examples from Mesopotamia.*

Algorithms and procedures are generally explained in cuneiform mathematical texts in a quite terse manner. Most often, the modalities of reasoning underlying them can be detected only through indirect evidence, such as terminology, structures of texts, diagrams, layouts, tabular formats, marginal notations, or epigraphy. A landmark work on evidence provided by terminology has been accomplished by Jens Høyrup (2002).

This presentation proposes a reflexion on the way in which other elements of reasoning can be grasped through the observation of diagrams and layouts. Adeline Reynaud will present some mathematical problems in which key elements of the solving procedure are absent from the discursive text and suggested only by a diagram drawn beneath it. She will analyse how these visual elements are articulated to those provided by means of words, and what this may reveal on the (non-)deductive character of the procedures. Christine Proust will present numerical texts that do not exhibit any discursive element. She will also analyze how the arrangement of the numbers on the surface of a clay tablet reflects the implementation and justification of an algorithm.

- !!!
**Changement d’horaire 14:00**Alan Baker (Swarthmore College, Swarthmore, PA)*Induction, Explanation, and Applied Mathematics*Some mathematical proofs are explanatory and others are not. I shall refer to a true, general mathematical proposition that lacks an explanatory proof as a “mathematical accident.” My interest in this talk is how mathematical accidents function in the context of applied mathematics. I shall focus on two main questions. Firstly, can mathematical accidents be used to explain scientific phenomena? Secondly, can the use of mathematical accidents in an applied context provide them with non-deductive support?

Scientific Directors : Pr. Michael Detlefsen (University of Notre Dame) et Pr. Jean-Jacques Szczeciniarz (Université de Paris 7-Diderot)

Sponsors: Pics Proofs and Computations, & Bergische Universität Wuppertal.

Also in this section :

- Science and Philosophy from Ancient to Modern Age 2017–2018
- Powers of Imagination. Historical Approach. 2017–2018
- History Review of the technical modernity in France (the twentieth century to the present) 2017–2018
- Meetings HPS of Paris Diderot 2017–2018
- AXIS HISTORY AND PHILOSOPHY OF MATHEMATICS 2017–2018
- History and Philosophy of Mathematics 2017–2018
- Mathematics from Ancient to Modern Age 2017–2018
- Reading Mathematical Texts 2017–2018
- Mathematics 19th and 20eth, History and Philosophy 2017–2018
- AXIS HISTORY AND PHILOSOPHY OF SCIENCE OF NATURE<BR>
- History and Philosophy of Physics 2017–2018
- Averroes’s Cosmology: The
*Middle Commentary*on the*De Caelo*of Aristotle 2017–2018 - History and Philosophy of Chemistry 2017–2018
- Psychophysical entanglement 2017–2018
- AXIS HISTORY OF PHILOSOPHY FROM ANCIENT TO MODERN AGE
- The Aristotle Metaphysics of Alexander of Aphrodisias 2017–2018
- The intentional and the real 2017–2018
- Memoria 2017–2018
- Power and Acting in Averroes: the
*Great Commentary*to*Metaphysics*Theta 2017–2018 - AXIS INTERDISCIPLINARITY IN HISTORY AND PHILOSOPHY OF SCIENCE